Characterization of symmetric polyconvexity in higher dimensions

In this work, we aim to characterize symmetric polyconvex functions in higher dimensions. Building on the characterization provided by Boussaid et al., for 2d and 3d cases, we extend their technique to higher dimensions to provide a characterization of symmetric polyconvex functions. Our main result shows that a function is symmetric polyconvex if and only if it can be expressed as a convex function of the matrix and its second-order minors, having a non-increasing behavior with respect to the variable of on second-order minors. The concept of S-positive semi-definite matrices is also introduced and analyzed and used as a main ingredient in the characterization. Moreover, This characterization allow us to identify the class of symmetric polyconvex quadratic forms, and show that there is no non trivial symmetric poly-affine functions.